Then you basicly declared morality a constant value within the laws of the universe. That's an axiom. Makes some pretty dull conclusions, but fundementally, not worse.

It's not about how strongly you feel about them, it's simply the fact that you would be unable to justify them any further that makes them an axiom. Mathematicians don't feel anything, at all, about their axioms - just say that in the context that the following are true, and make their conclusions there on out. Note also that an axiom doesn't necessarily have to be true (or rather, they are automaticaly true, by their own definition) in any context, just non-contradictory with other axioms. They also don't have to be complete, it's fine if they can't resolve some problems (note that it's technically impossible for a set of axioms to be both complete and coherent.) 

I use a cheap cop-out, only 1, fairly weak, axiom. No contradictions, but really limited capacity to evaluate situations.

I'd be interested in hearing what these not-axioms you carry around are, though.

Some people are evil and don't deserve to live

Be Excellent to One Another

You Are What You Choose to Be

Nobody deserves forgiveness, but they may still have the choice to receive it.

There is no greater good. There is a such thing as picking the least evil option, but there's still no greater good.

Life sucks. So what?

There is no subjective reality. But there are a lot of people who don't realize that what they do matters.

A true man never dies, even when he is killed.

Beware of situations where it's Us vs Them. You might not be on the side you thought you were.

Never Stop Running. Away from it, Towards it, Regardless of it... that's not the focus here.

I'm sorry, I have a hard time following this. I have a few core beliefs, but whether or not they classify as an axiom, I'm not sure of myself.

I there are any beliefs that are not justified by another belief, (which should be the case, if your logic isn't circular), they are axioms. As said, the test is asking "why do I believe that" and if the only answer that you can give is that it seems so blantantly obvious, then it's an axiom.

Axioms are important, and annoying, because if 2 people with different axioms start debating, they can eventually come to a point where they are both, in their own axiomatic system, perfectly coherent, but come to different conclusions as they start with different axioms; making any further discussion 100% pointless.

I know what an axiom is, and I'm glad we're on the same page about them. I just don't believe it's even possible to get to such personal axioms you're suggesting because of human nature unless you accept that those axioms can change - in which case they probably can't even be called axioms. Anything I might consider an axiom at any point of time during my life could change later on because of a change in thinking. For example, one might think that no person has the right to take another's life, but it might not be that difficult to sway a person's opinion to be more positive about death sentence in regard to certain serious crimes. It's possible that certain thoughts can be reduced to simple axioms, but at the same time, there's probably a humongous amount of things that can only be reduced to strong beliefs that are still subject to change.

I strongly believe in human rights, yet it's difficult to define human rights precisely. Some human rights are commonly accepted, but some are more questionable and it's hard to say whether they should be considered human rights - or where to draw the limits. For example, I consider privacy to be more or less a human right, but I'm not sure I can define to what extent. Obviously it can't be defined to apply fully in all cases (for example criminal investigation), but where is the line then? I'm not sure if liberty is considered a human right, but it's something I value highly that suffers from the same problem as privacy: To what extent can it be expected? To me, it's all a huge mess that's about balancing the different aspects so that no single aspect can dominate the rest just to keep the whole system self-consistent.

If I had even one axiom, it could potentially endanger other things I find valuable just to keep things consistent with the axiom. However, if I can simply change the axiom if it causes problems, how can you even call it an axiom? Thus, it seems that if there's even one axiom, there must be a whole system of them, which every single view a person can have can be derived from. I'm not convinced you can, in practice, come up with such a comprehensive system, especially in a way that doesn't contradict itself. I wouldn't be surprised if, given any two personal 'axioms', it would be possible to come up with a contradiction between them. I'm not saying that's the case, but I wouldn't be surprised if it was.

Video footage is the best witness.

I like turtles and boobs. I don't like boobs on a turtle.

Yeah.... what? How is there a right and wrong when no absolute truths exist?

Also none of these virtues sound even close to our current moral understanding.